Volume 10 (2003), Number 1, 37–43
BRANCHED COVERINGS AND MINIMAL FREE RESOLUTION FOR INFINITE-DIMENSIONAL COMPLEX
SPACES
E. BALLICO
Abstract. We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V) (e.g. complete intersections or cones over
finite-dimensional projective spaces). In the former case we obtain the vanishing result forH1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem forP(V).
2000 Mathematics Subject Classification: 32K05, 32L10.
Key words and phrases: Infinite-dimensional projective space, complex Banach manifold, branched covering, vanishing of cohomology.
1. Minimal Free Resolution
Recently, L. Lempert (see [7], [8], [6], and the references therein) gave a new life to infinite-dimensional holomorphy. The aim of this paper is to give an algebraic set-up in which these general results may be applied. We found two different set-ups and to each set-up is devoted one of the two sections of this paper. For general results on analytic sets in locally convex spaces and the projectivization of a Banach space, see [9] and [10].
In the second section we will consider nice branched coverings f :Y →M of a complex manifoldM. For certain particular ramified coverings of an infinite-dimensional projective space and a fine study of their Dolbeaut cohomology, see [6, Ch. 4].
In the first section of this paper we consider finite codimensional closed ana-lytic subspaces, Y, of a nice projective space P(V) such that the ideal sheaf IY
ofY inP(V) admits a minimal free resolution of finite length. Since the results on Dolbeaut or algebraic cohomology for P(V) are proved in various degree of generality concerning the locally convex space V and the corresponding results for sheaf cohomology are just conjectural, we will just take as axiom thatV sat-isfies the conditions needed to have [7, Theorem 7.3 (i)] for sheaf cohomology, not just Dolbeaut cohomology.
Definition 1. LetV be an infinite-dimensional complex vector space equipped with a locally convex Hausdorff topology. We will say thatV has Property (A) or that it satisfies Condition (A) if every holomorphic line bundle on P(V) is a multiple of the hyperplane line bundle OP(V)(1) andHi(P(V), L) = 0 for every
holomorphic line bundleLonP(V) and every integeri >0. We will say that V
has Property (B) or that it satisfies Condition (B) ifHi(P(V), L) = 0 for every
holomorphic line bundle L onP(V) and every integer i > 0. We will say that
V has Property (C) or that it satisfies Condition (C) ifHi(P(V),O
P(V)(t)) = 0
for all integers t,i such that i >0.
Unless otherwise stated, we will always consider sheaf cohomology groups in Definition 1 and elsewhere in this paper. When the base space is paracompact, these cohomology groups are isomorphic to the ˇCech cohomology groups of the same sheaf [4, Corollary at p. 227]. We recall that for every Banach spaceV the projective space P(V) is metrizable. For instance, this can be checked defining on P(V) the so-called Fubini–Study metric or, calling S the unit sphere of V, using that the restriction to S of the quotient map π : V\{0} → P(V) has compact fibers to define a metric onP(V) from the norm metric onS. Thus by a theorem of Nagata and SmirnovP(V) is paracompact. IfV has no continuous norm, then P(V) is not even a regular topological space [7, p. 507].
Definition 2. LetV be an infinite-dimensional complex vector space satisfy-ing Condition (C) andY a closed analytic subspace of P(V). We will say that
Y has finite cohomological codimension in P(V) if the ideal sheaf IY of Y in
P(V) has a finite free resolution, i.e., if there are an integer r≥0 and sheaves
Ei, 0≤i≤r, on P(V) with each Ei a direct sum of finitely many line bundles OP(V)(t), t∈Z, such that there there is an exact sequence
0→Er →Er−1 → · · · →E1 →E0 → IY →0. (1)
Every mapEi →Ei−1 in (1) is given by a matrix of continuous homogeneous
polynomials. We will say that (1) is minimal if for every integeriwith 1 ≤i≤r
the matrix associated to the map Ei →Ei−1 has no entry which is a non-zero
constant.
As in the case of a finite-dimensional complex projective space a minimal free resolution is essentially unique, i.e. for every integer the integer r and the isomorphism classes of all vector bundlesEi, 0≤i≤r, are uniquely determined
byY. Furthermore, from any finite free resolution it is easy to obtain a minimal free resolution, just deleting certain direct factors of the bundles Ei. The big
difference is that in the infinite-dimensional case we were forced to assume the existence of a free resolution of IY (and we believe that this is a restrictive
assumption), while every closed analytic subspace of Pn has a free resolution because it is an algebraic subscheme (Chow theorem) and we may apply Hilbert syzygy Theorem. Only the Chow theorem is available for an arbitrary finite-codimensional closed analytic subset Y of the projectivization of an infinite-dimensional Banach space [10, III.3.2.1].
Remark 1. We only know two classes of subvarieties Y ⊂ P(V) with finite free resolution. The first class is given by the complete intersections of r+ 1 continuous homogeneous polynomials. In this case a minimal free resolution of IY is the Koszul complex of r+ 1 forms. Here is the second class. Take
Z ⊆ P(A) be any projective variety. Let Y ⊆ P(V) be the cone with vertex P(W) and base Z. Since dim(A) is finite, Z has a minimal free resolution as a subscheme of P(A). Every homogeneous form on A induces a continuous homogeneous form on V not depending on the variable W. Hence a minimal free resolution of Z in P(V) induces a minimal free resolution of Y in P(V) with the same numerical invariants (the integer r, ranks of the bundles Ei and
degrees of their rank one factors).
Theorem 1. LetV be an infinite-dimensional complex vector space satisfying Condition (C) and Y a closed analytic subspace of P(V) with finite cohomo-logical codimension. Then Hi(P(V),I
Y(t)) = 0 and Hi(Y,OY(t)) = 0 for all
integers t, i such that i >0.
Proof. Fix a finite free resolution (1) of IY and twist it with OP(V)(t). By
Condition (C) we have Hi(P(V), E
j(t)) = 0 for 0 ≤ j ≤ r and every integer
i > 0. Hence the vanishing of Hi(P(V),I
Y(t)) follows a splitting of (1) into
short exact sequences. From the exact sequence
0→ IY(t)→ OP(V)(t)→ OY(t)→0 (2)
and the vanishing of Hi(P(V),O
P(V)(t)) (Condition (C)) for every i > 0 we
obtain Hi(Y,O
Y(t)) =Hi+1(P(V),IY(t)) = 0. ¤
Corollary 1. LetV be an infinite-dimensional complex vector space satisfying Condition (C) and Y a closed analytic subspace of P(V) with finite cohomo-logical codimension. Assume P(V) paracompact. Then the topological and the holomorphic classification of line bundles on Y are the same.
Proof. The sheaf cohomology groupH1(Y,O∗
Y) classifies isomorphism classes of
all line bundles on any reduced complex space Y. Thus H1(P(V),O∗
P(V))∼=Z.
From the exponential sequence
0→Z→ OP(V) → OP∗(V) →1 (3)
and Theorem 1 we obtainH(Y,O∗
Y)∼=H2(Y,Z). The corresponding exponential
exact sequence for topological line bundles and the paracompactness assumption imply thatH2(Y,Z) parametrizes equivalence classes of topological line bundles
on Y. Since these isomorphisms are compatible with the map associating the topological structure to any holomorphic line bundle, we conclude that the
assertion is valid. ¤
2. Branched Coverings
Definition 3. LetY be a connected complex analytic space,M a connected complex manifold andf :Y →M a holomorphic map. We say thatf is ac-flat finite covering if it is a proper map with finite fibers andf∗(OY) is a locally free OM-sheaf with finite rank. The rank off∗(OY) is called the degree deg(f) off.
on Y and the associated inclusionOM →f∗(OY) has as cokernel a locally free OY-sheaf of rank deg(f)−1.
Example 1. LetV be an infinite-dimensional Banach space such thatP(V) is localizing ([7, p. 509]), Y a connected analytic space and f : Y → P(V) a
c-flat finite covering of degreed. By [7, Theorems 7.1 and 8.5], there are integers
ai, 1≤ i≤d−1, such that a1 ≥ · · · ≥ad−1 and f∗(OY)/OP(V) ∼=OP(V)(a1)⊕ · · · ⊕ OP(V)(ad−1). The restriction of f∗(OY)/OP(V) to any line shows that the
(d−1)-ple (a1, . . . , ad−1) is uniquely determined byf. Following the terminology
used in the theory of Riemann surfaces we say that (a1, . . . , ad−1) are the scrollar
invariants off. We haveH0(Y,O
Y) = Cif and only ifa1 <0. For every integer
t <−a1 the natural map f∗
:H0(P(V),O
P(V)(t))→H0(Y, f∗(OP(V)(t))) is an
isomorphism. The natural map
f∗
:H0(P(V),OP(V)(−a1))→H0(Y, f∗(OP(V)(−a1)))
is not surjective and this is a characterization of the integer a1.
Remark 2. Let f :Y →M be a c-flat finite covering of degree d. Let F be a locally freeOY-sheaf on Y with finite rankr andP ∈M. For eachQ∈f−1(P)
there is a neighborhood UQ of Q such that F|UQ ∼= U ⊕r
Q . Since f
−1(P) is
finite,f is proper and Y is Hausdorff, we can find such neighborhoods UQ with
UQ∩UQ′ = ∅ for all Q, Q′ ∈ f−1(P) with Q 6=Q′ and a neighborhood U of P
such thatf−1(U) is the disjoint union of such open setsU
Q, Q∈f−1(P). Since
f∗(OY) is locally free of rank d, we see that f∗(F) is locally free of rank rd.
Hence ifV is a Banach space such thatP(V) is localizing andM =P(V), then
f∗(F)∼= Lrd
i=1OP(V)(ti)) for some integers ti [7, Theorems 7.1 and 8.5]. Hence
as in the previous example F is uniquely determined by rdintegers.
Example 2. Let f : Y → M be a proper holomorphic map with finite fibers between finite-dimensional complex spaces. Assume M smooth. The map f is c-flat if and only Y is locally Cohen–Macaulay, i.e., if and only if for every P ∈ Y the Noetherian local ring OY,P is Cohen–Macaulay [3, Example
III.10.9]. If Y is smooth, then it is locally Cohen–Macaulay [3, p. 184]. Then we can obtain in an easy way several infinite-dimensional examples taking an infinite-dimensional complex space Z and taking M′
:= M ×Z, Y′
:=Y ×Z
and f′
:Y′
→M′
induced by f.
Remark 3. (a) The composition of two c-flat maps isc-flat.
(b) A locally biholomorphic and proper map between two complex manifolds is c-flat.
Remark 4. Let f : Y → M be a proper map with finite fibers and F any sheaf on Y. By [5, Ch. 1, Theorem 4], we have Rfi
∗(F) = 0 for every i >0.
Corollary 2. Fix an integer t ≥1. Let f : Y →M be a c-flat map between complex spaces. Assume that for every finite rank holomorphic vector bundle
E on M we have Ht(M, E) = 0. Then Ht(Y, F) = 0 for every finite rank
holomorphic vector bundle F on Y.
Proof. By Remark 4 we have Rfi
∗(F) = 0 for every i > 0. Thus H
t(Y, F) ∼=
Ht(M, f
locally free sheaf on M with finite rank. Hence Ht(M, f
∗(F)) = 0 by our
assumption on M. ¤
Example 3. Fix an integer d ≥ 2, a complex space M, a holomorphic line bundleL onM and an effective Cartier divisor D onM such that OM(−D)∼=
L⊗d; we require that L be locally holomorphically trivial, i.e. our definition of
line bundle be more restrictive than the one in [7, p. 490]. By [1, Example 1.1], these data are sufficient to construct a complex space Y and a proper holomorphic map f : Y → M such that f∗(OY) ∼= OM ⊕L⊕ · · · ⊕L⊗(d−1).
Hence f∗(OY) is locally free. Thus f is c-flat if M is smooth. This is the
classical construction of a cyclic branched covering with D as a ramification divisor. One can see Y as a closed analytic subset of the total space of the line bundle L∗
and f is the restriction of the natural mapL∗
→M toY. As in [1, Example 1.1], one can even give local equations forY insideL∗
in terms of local equations of D insideM. In particular, Y is isomorphic to an effective Cartier divisor ofL∗
. From these local equations it follows that ifM andDare smooth, then Y are smooth. Fix an integert >0. By Remark 4 and the Leray spectral sequence of f we obtain Ht(Y,O
Y)∼= Ld−1
i=0 H
t(M, L⊗i). Hence Ht(Y,O Y) = 0
if and only if Ht(M, L⊗i) = 0 for every integer i with 0≤i≤d−1.
The vanishing theorem for Dolbeaut cohomology groups corresponding to the next result is proved in [7, Theorem 7.3].
Proposition 1. LetV be a complex Banach space. Then the sheaf cohomology group H1(P(V),O
P(V)(x)) vanishes for every x≤0.
Proof. By [7, Theorem 7.1], for every holomorphic line bundleLonP(V) there is an integert such thatL∼=OP(V)(t). The sheaf cohomology groupH1(Y,OY∗)
classifies the isomorphism classes of all line bundles on any reduced complex space Y. Thus H1(P(V),O∗
P(V))∼=Z.
Claim: We have H1(P(V),Z) = 0.
Proof of the Claim: It would be easy to check that the first singular co-homology group vanishes just by checking that P(V) is simply-connected to obtainH0(P(V),Z) = Z,H1(P(V),Z) = 0 and then by applying the universal-coefficient theorem [12, p. 243]. But for sheaf cohomology we follow a longer path. Since V and V\{0} are paracompact, sheaf cohomology and sheaf ˇCech cohomology on V and V\{0} are the same [4, p. 228]. By [12, p. 334], sheaf
ˇ
Cech cohomology and Alexander cohomology agree on V and V\{0}. By [11, Corollary 1.3], the inclusion V\{0} →V is a homotopy equivalence. V is con-tractible. Alexander cohomology satisfies the homotopy axiom [12, p. 314]. Hence H1(V\{0},Z) = 0 (sheaf cohomology). Let π : V\{0} → P(V) be the
quotient map. Thusπis a locally trivial fibration withC∗
as fiber. The spectral sequence of π induces an exact sequence for sheaf cohomology
0→H1(P(V), π∗(Z))→H1(V\{0},Z)→H0(P(V), π∗(Z)) (4)
(a part of the five term exact sequence arising from any first quadrant spectral sequence). ThusH1(P(V), π
to prove that π∗(Z) ∼= Z. Notice that π∗(Z) is locally isomorphic to the
con-stant sheaf Z. To check that it is globally isomorphic to Z one can either use the simply-connectedness of P(V) (to prove it use the simply-connectedness of
V\{0} just proven and the long exact sequence of homotopy groups associated to the fibrationπ [13]) or the fact that it has a global nowhere vanishing section because H0(P(V), π
∗(Z)) = H0(V\{0},Z) = Z. Hence the proof of the Claim
is finished.
From the Claim and the exponential sequence (2) we obtain
H1(P(V),OP(V)) = 0.
Now assume x <0. There is a reduced hypersurface T of P(V) with deg(T) =
−x. Hence H0(T,O
T) = C. Thus the restriction map H0(P(V),OP(V)) → H0(T,O
T) is surjective. From the exact sequence
0→ OP(V)(x)→ OP(V) → OT →0 (5)
we obtainH1(P(V),O
P(V)(x)) = 0. ¤
The referee made the following very useful remark.
Remark 5 (due to the referee). Let V be an infinite-dimensional Banach space such thatP(V) admits smooth partitions of unity. Then for every integer
x the ˇCech cohomology group ˇH1(P(V),O
P(V)(x)) is naturally embedded into
the Dolbeaut group H0,1(P(V),O
P(V)(x)) which vanishes by [6, Theorem 7.3].
Hence if P(V) admits smooth partitions of unity, then Proposition 1 is true also for every positive integer x. V admits smooth partitions of unity if it is a separable Hilbert space; for more examples see [7, p. 511], or [2, §8], and the references therein.
Remark 6. Let V be an infinite-dimensional complex Banach space. Fix integers d, y such that d ≥ 2 and y < 0. Let D be a closed and reduced hypersurface of P(V) with deg(D) = −yd. Make the construction of Ex-ample 3 taking Y = P(V) and L = OP(V)(y). By Proposition 1 we have H1(P(V),O
P(V)(iy)) = 0 for every integer i such that 0 ≤ i ≤ d−1. Hence H1(Y,O
Y) = 0. Thus by the exponential sequence (2) every topologically
trivial holomorphic line bundle on Y is holomorphically trivial. Now fix an integer x < 0 and set R := f∗
(OP(V)(x)) ∈ Pic(Y). For every locally free
sheaf A onP(V) we have f∗(f ∗
(A))∼=f∗(OY)⊗A (projection formula). Thus
f∗(R)∼= Ld−1
i=0 OP(V)(x+iy). Hence by Proposition 2, Remark 4 and the Leray
spectral sequence of f we obtain H1(Y, R) = 0.
Acknowledgement
I want to thank the referee for several very useful remarks.
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(Received 14.01.2002; revised 24.07.2002)
Author’s address: Dept. of Mathematics University of Trento 38050 Povo (TN) Italy
